Texture uv map is not applied to the object while I'm following the projection paiting tutorial
I think there is a little misunderstanding. 

At point 1, you are talking about a texture. But at point 2, you are in the material panel. So you wont find your texture here.

An object is rendered with one (or more) material(s). And a material may use an image to render.

As you are using Cycles rendering :


Go to the material tab
Open the "surface" section
Click on the little dot near the color
Choose "Image texture"
Choose or open your image here (you'll have to choose your image by the image's name, not the texture's name : Cycles uses images directly)